This is my problem in my textbook. In my thinking, $\operatorname{rank}T$ is equal to rank of $(p(x)) = 3$. But, I think it 's wrong, because I haven't used $p(x+1)$ yet. At b, I really don't know how to find.

$P_2$ is a three dimensional vector space That is to say it is finite dimensional space. And $T$ is a linear transformation. So $T$ has a matrix representation ! It is a basic fact of linear algebra. Anyway $x^2$, $x$ and 1 is corresponded to the vectors (0,0,1), (0,1,0), (1,0,0) For instance $x+1$ is corresponded to (1,1,0). And $T(x^2)=(x+1)^2$ so that $T$ sends (0,0,1) to (1, 2, 1).
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HK LeeNov 9 '12 at 13:37